Combinational Circuit
Definition
A Combinational Circuit can provide output instantly based on the current input. It only listens for input signal and and generates output no matter what is the past input as it has no feedback or memory component.
Based on this property of caring only about the present input, we can use the idea of function in maths to vividly understand combinational circuits.
Combinational circuits are functions
In the figure below, the combinational circuit actually acts as a function . Given the same input (input signal) in whatever order, it will always generate the same output (output signal).
Combinational Design Process
Capture the function
Form the truth table according to the given behavior, and write down the SOm or POM.
Form the equation (SOm or POM) directly by counting all the possible desired cases.
Convert to circuit
Create equation: First, if you used a truth table in Step 1, create equations
Implement using gates: For each output (term in the equation above), create a circuit corresponding to that output's equation.
Example: Three 1s Pattern Detector
Problem Statement (Behavior description): Detect three adjacent 1's in an 8-bit input. For example, 00011101 -> 1, 10110011 -> 0, 01110001 -> 1, etc.
Capture the function
There are 8 inputs here, and it will be too tedious if we create a truth table. So, we write the SOm directly, and it will be
where is the output, describes the eight inputs
Convert to Circuit
Create equation: We have already create equation in step 1, and it is obviously not the simplest standard form. But we still can use the expression to implement the gate, it's just that it will be a bit tedious.
Implement using gates: For each output term, like , we will use a AND gate with three inputs and one output (denoted as abc on the output line). The final gate level implementation should look as follows
Decoder
A decoder is a basic circuit that takes a binary input number, and outputs a corresponding one-hot output.
Attention
"One-hot" means only one bit of the output is high, its position corresponds to the input value.
Output width is always 2 to the power of input width. e.g., N-input decoder has outputs.
In general, decoders can be referred to as n-m decoders. e.g., 2-4 decoder, 3-8 decoder, etc.
Internal Design
Based on the property of the decoder, to design a decoder, we can use AND gate for each output to detect input combination. e.g., the following is the internal design for a 2-4 decoder.
Example: 7-Segment Decoder
A seven segment decoder is a basic circuit that maps a 4-bit binary input to a 7-bit binary output, where in the output, multiple bits can be set to HIGH, and each bit is connected to one LED.
Implement a 7-Segment Decoder
Capture the Function
Here, we form the truth table for the 7-segment decoder.
0
0000
1
1
1
1
1
1
0
1111110
1
0001
0
1
1
0
0
0
0
0110000
2
0010
1
1
0
1
1
0
1
1101101
3
0011
1
1
1
1
0
0
1
1111001
4
0100
0
1
1
0
0
1
1
0110011
5
0101
1
0
1
1
0
1
1
1011011
6
0110
1
0
1
1
1
1
1
1011111
7
0111
1
1
1
0
0
0
0
1110000
8
1000
1
1
1
1
1
1
1
1111111
9
1001
1
1
1
1
0
1
1
1111011
Inputs 1010 to 1111 are invalid BCD and can be treated as "don't care" or produce blank/error display depending on the design.
Convert to Circuit
Create equation: Here, we need to create a circuit for each segment, thus we should generate a K-map for each segment/output a,b,c,d,e,f,g, and then obtain the Boolean Expression for each segment/output.
Implement using gates: Nothing tricky here. Just use gates to implement out the Boolean expression for each segment.
Notes
In combinational circuit, each output bit has its own logic circuit
In this 7-segment decoder example, we notice that each segment/output bit has its own boolean expression, a.k.a, has its own logic circuit. This is actually a very common design in combinational circuit, where each output bit is designed independently, using its own Boolean expression (derived from truth tables or K-maps).
Multiplexer
A multiplexer (MUX) is a device that selects one of several input signals and forwards it to a single output line. Thus, a multiplexer will have
Several input signal lines
One or several select lines
"One" output signal line
Attention
An input MUX requires a -bit select.
"One" output doesn't exactly mean one-bit output, it depends on how you treat "one". e.g., one can represent one-bit, and it can also represent one pack of several bits. You will see this in the later example.
Internal Design
The internal design of a multiplexer also uses an AND gate to "filter" each input signal according to the select line input and then combine them together using a OR gate.
Example: Use multiplexer to select multiple bits
This is done by combining several multiplexers together. e.g., the following is a 4bit 2x1 multiplexer, which means the two inputs are two 4-bit signal, and the one select line will select which 4-bit input will be forwarded to the output. (This is the meaning behind the "One" from above)
In the implementation, it is actually done by combining four 2x1 multiplexers which shares the same select line to select between A and B.
Notes
Multiplexer Drawing Convention
MUXes are so common that they are often drawn using their own symbol.
Timing Diagram
Definition
Timing diagram is a diagram that shows the behavior of a circuit with progression of time.
Example: AND Gate
The timing diagram for an AND gate is shown as follows,
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